Solving Algebraic Equations Flashcards

An electrical apprentice is told that the total resistance (R_T) of two resistors in parallel is given by the formula 1/R_T = 1/R_1 + 1/R_2. If R_T is 6 ohms and R_1 is 10 ohms, what is the value of R_2?

Answer: 15 ohms

To solve for R_2, substitute the given values into the equation: 1/6 = 1/10 + 1/R_2. To isolate 1/R_2, subtract 1/10 from both sides: 1/6 - 1/10 = 1/R_2. Find a common denominator (30): 5/30 - 3/30 = 1/R_2. This simplifies to 2/30 = 1/R_2, or 1/15 = 1/R_2. Therefore, R_2 = 15 ohms.

Solve the following linear equation for x: 5(x + 3) - 2x = 21

Answer: x = 2

First, use the distributive property to eliminate the parentheses: 5x + 15 - 2x = 21. Next, combine like terms on the left side of the equation: 3x + 15 = 21. Then, subtract 15 from both sides to isolate the term with x: 3x = 6. Finally, divide both sides by 3 to solve for x: x = 2.

A job requires cutting a 42-foot length of conduit into two pieces. One piece must be 8 feet longer than the other. Which of the following equations correctly represents this situation, where 'L' is the length of the longer piece?

Answer: L + (L - 8) = 42

Let 'L' be the length of the longer piece. The shorter piece is 8 feet shorter than the longer one, so its length can be represented as 'L - 8'. The sum of the lengths of the two pieces is 42 feet. Therefore, the equation is L + (L - 8) = 42.

If 3a - 5 = 16, what is the value of 6a + 10?

Answer: 52

First, solve the initial equation for 'a'. Add 5 to both sides: 3a = 21. Divide by 3: a = 7. Now, substitute the value of 'a' into the second expression: 6(7) + 10. This equals 42 + 10, which is 52.

Solve the following system of equations for y: 2x + y = 11 3x - 2y = 6

Answer: y = 3

One way to solve this is using the substitution method. First, solve the top equation for y: y = 11 - 2x. Now, substitute this expression for y into the second equation: 3x - 2(11 - 2x) = 6. Distribute the -2: 3x - 22 + 4x = 6. Combine like terms: 7x - 22 = 6. Add 22 to both sides: 7x = 28. Solve for x: x = 4. Finally, substitute x = 4 back into the first equation: 2(4) + y = 11, which simplifies to 8 + y = 11. Solving for y gives y = 3.

Which of the following is a correct factorization of the quadratic expression x² - 2x - 15?

Answer: (x + 3)(x - 5)

To factor the quadratic expression x² - 2x - 15, we need to find two numbers that multiply to -15 and add to -2. The pairs of factors for -15 are (1, -15), (-1, 15), (3, -5), and (-3, 5). The pair that adds up to -2 is +3 and -5. Therefore, the factored form is (x + 3)(x - 5).